Fixed Point Theorems in B-Metric Spaces and Their Applications To Non-Linear Fractional Differential and Integral Equations

Abstract

In this paper, we modify L-cyclic (alpha, beta)(s)-contractions and using this contraction, we prove fixed point theorems in the setting of b-metric spaces. As an application, we discuss the existence of a unique solution to non-linear fractional differential equation, D-c(sigma) (x(t)) = f(t, x(t)), for all t is an element of (0, 1), (1) with the integral boundary conditions, x(0) = 0, x(1) = integral(rho)(0) x(r) dr, for all rho is an element of (0, 1), where x is an element of C([0, 1], R), D-c(alpha) denotes the Caputo fractional derivative of order sigma is an element of (1, 2], f : [0, 1] x R -> R is a continuous function. Furthermore, we established existence result of a unique common solution to the system of non-linear quadratic integral equations, {x(t) = integral(1)(0) H(t, tau)f(1)(tau, x(tau))d tau, for all t is an element of [0, 1]; x(t) = integral(1)(0) H(t, tau)f(2)(tau, x(tau))d tau, for all t is an element of [0, 1], where H : [0, 1] x [0, 1] -> [0, infinity) is continuous at t is an element of [0, 1] for every tau is an element of [0, 1] and measurable at tau is an element of [0, 1] for every t is an element of [0, 1] and f(1), f(2) : [0, 1] x R -> [0, infinity) are continuous functions.